yes, you should definitely wash it off. they are usually prewashed but can still have traces of saponnin- a kind of soapy residue. Just put them in a colander or strainer that's too small to let the grains through and run it under cold water. It's not total mush when it's cooked correctly and it's still a little crunchy. you can tell when it's done cooking because the grains become translucent, and the white germ partially detaches itself and ends up looking like a white-spiraled tail.
Yes, do rinse it. It's always a good idea to rinse any grains or beans (yeah, I know it isn't REALLY a grain), but if you don't rinse quinoa well you'll get a soapy taste.
The texture totally depends on how you cook it...cook it the way you like it. I've eaten it mushy, tender, and sort of crunchewy, depending on how long the person making it cooked it. I don't think any way is right or wrong, it just depends on what you personally like. I like it a bit more tender, but not mushy. I tend to eat it dry though, not with sauce, so that may make a difference.